Dualizing Initial Algebras

Neil Ghani1, Christoph Lüth2, Federico de Marchi1, John Power3
1Department of Mathematics and Computer Science, University of Leicester
2FB3 - Mathematik und Informatik, Universität Bremen
3Laboratory for Foundations of Computer Science, University of Edinburgh

Whilst the relationship between initial algebras and monads is well-understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial algebras form both monads and comonads.

In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by defining a notion cosignature and coequation and then proving the models of this syntax are precisely the coalgebras of the representing comonad.

Christoph Lüth, 02.05.2002